AP Calculus AB Exam Question - Product and Quotient Rules

Consider the function f(x) = (2x^3 + 4x^2) / (x - 1). Using the product and quotient rules, find the derivative of f(x).

Step-by-step Solution:

To find the derivative of f(x), we will use the product rule and quotient rule. Let's calculate it step-by-step:

Step 1: Apply the product rule First, let's differentiate the numerator and denominator separately:

Numerator (u): 2x^3 + 4x^2 u' = 6x^2 + 8x

Denominator (v): x - 1 v' = 1 (as the derivative of x is 1)

Step 2: Apply the product rule The formula for the derivative of a quotient is: (fg)' = (f'g - fg') / g^2

Now, let's apply the product rule to find the derivative of f(x):

f'(x) = [(u'v - uv') / v^2]

=> [(6x^2 + 8x)(x - 1) - (2x^3 + 4x^2)(1)] / (x - 1)^2

Expanding and simplifying further:

=> [6x^3 - 6x^2 + 8x^2 - 8x - 2x^3 - 4x^2] / (x - 1)^2

Simplifying again:

=> [4x^3 - 2x^2 - 8x] / (x - 1)^2

=> 2x(2x^2 - x - 4) / (x - 1)^2

Therefore, the derivative of f(x) is:

f'(x) = 2x(2x^2 - x - 4) / (x - 1)^2

This is the final answer, which represents the derivative of the given function using the product and quotient rules.